We present observations of the 85.69 GHz continuum emission and H42α line emission from the central 30 arcsec within NGC 4945. Both sources of emission originate from nearly identical structures that can be modelled as exponential discs with scalelengths of ∼2.1 arcsec (or ∼40 pc). An analysis of the spectral energy distribution based on combining these data with archival data imply that 84 ± 10 per cent of the 85.69 GHz continuum emission originates from free–free emission. The electron temperature is 5400 ± 600 K, which is comparable to what has been measured near the centre of the Milky Way Galaxy. The star formation rate (SFR) based on the H42α and 85.69 GHz free–free emission (and using a distance of 3.8 Mpc) is 4.35 ± 0.25 M⊙ yr−1. This is consistent with the SFR from the total infrared flux and with previous measurements based on recombination line emission, and it is within a factor of ∼2 of SFRs derived from radio data. The Spitzer Space Telescope 24 μm data and Wide-field Infrared Survey Explorer 22 μm data yield SFRs ∼10× lower than the Atacama Large Millimeter/submillimeter Array measurements, most likely because the mid-infrared data are strongly affected by dust attenuation equivalent to AV = 150. These results indicate that SFRs based on mid-infrared emission may be highly inaccurate for dusty, compact circumnuclear starbursts.